Simulation Study for Parametric EWMA and NPWEWPA-SR Control Charts Against Non-Normality Assumptions

Common control chart types such as EWMA require assumptions to have valid information. A simulation was conducted to compare IC robustness and OOC performance for parametric EWMA and NPEWMA-SR control charts in the presence of a violation of symmetrical assumption. Monte Carlo simulation study held scale parameters with various shape parameters in Weibull distribution. It was found parametric EWMA and NPEWMA-SR control charts were unsuitable for applying asymmetrical distribution due to weak IC robustness and frequent false alarm. Although EWMA-X ̅ The control chart showed a most stable OOC performance; the weak IC robustness made the control chart unacceptable. Whereas, NPEWMA-SR control chart lost the ability in small shift detection when symmetrical assumption violated. Moreover, two different weightage of current sample for both parametric EWMA and NPEWMA-SR control charts were also investigated. The results showed that weightage of current sample for both parametric EWMA and NPEWMA-SR control charts did not affect the ARL value trend in different skewness of Weibull distribution.


INTRODUCTION
Weibull distribution is a distribution with characteristics of many different types of distributions.This fact makes it famous among engineers and quality practitioners and is widely used in many fields such as survival analysis, various industrial areas and quality control.Weibull distribution contains two parameters: scale parameter, λ and shape parameter, k.The probability density function of Weibull distribution is written as follows: In the Equation (1), the k value is set to obtain the skewness coefficient value from 0.1 to 9.0 with fixed λ = 1 [1].The value of skewness is inversely proportional to the shape parameter, k and the value of skewness coefficient is calculated by the following expression (Equation ( 2)): Since the NPEWMA-SR control chart is modified by adding concept of Wilcoxon Signed-Rank statistic, the properties of the control chart is related to Wilcoxon Signed-Rank statistic.Consequently, brief information about Wilcoxon Signed-Rank statistic is introduced.The test is a non-parametric procedure as an alternative to the t-test statistic when comparing two related samples, matched samples, or a repeated measurements on a single sample [2].To compare the performance between non-parametric and parametric statistics, asymptotic relative efficiency (ARE) is used, ([3][4]).ARE is an efficiency comparison measurement where ARE value represents the efficiency ratio between two statistic procedures or tests.[5] and [6] stated that the ARE of signed-rank test relative to t-test is 0.955, 1, 1.097 and 1.5 for the Normal, Uniform, Logistic and Laplace distributions, respectively.The ARE values above indicate signed-rank test is only slightly weaker than t-test in Normal distribution, and signed-rank test is much powerful than t-test in nonnormal distribution.Consequently, the non-parametric statistical test is widely used in many sectors due to only few underlying assumptions need to be fulfilled, such as the samples must come from continuous and symmetrical distribution.[7]- [9] have shown that the Wilcoxon Signed-Rank test is not robust against the symmetrical assumption.
This study investigates the robustness of in-control (IC) and out-of-control (OOC) performance for parametric EWMA and NPEWMA-SR control charts in violation of symmetrical assumption using simulation study.Many research have been conducted to study the robustness of various statistical test, such as [10]- [12] .Meanwhile, [13] have studied the application of permutation theory to derive robustness criteria for the details regarding a statistical test's robustness.They mentioned two criteria for a good statistical test: quick detection in change of specific factor and insensitivity to detect change in extraneous factor that not include in the test.The first criterion is the test's power, whereas the second criterion is the robustness of the test.In this paper, the robustness of each control charts is used.

METHODS
NPEWMA-SR control chart is a non-parametric control chart, where although it s relatively robust to the normality assumption, but the requirement of symmetrical distribution should not be violated.Hence, the impact of violation in symmetrical assumption to control charts is studied through a simulation study.The simulation study in this study mainly aims to learn the robustness of parametric EWMA and NPEWMA-SR control charts against the assymmetrical assumption.
The first step is choosing the design parameters with approximately the same performance for each control chart to avoid unsuitable comparisons.The control chart's design parameter is divided into two parts.The first part is chosen based on best combination of (L, λ) on ARL0 500  and smallest ARL1 in detecting δ =1.5 shift of in the process mean in simulation study of Normal distribution.The second part is to use the same λ value with λ=0.10 for all control charts and choose the combination of ARL0 500  in simulation study of Normal distribution.The design parameters in the first part are EWMA (L=2.998,λ=0.25, n=1), EWMA- ̅ (L=3.1, λ=0.25, n=10) and NPEWMA-SR (L=2.905,λ=0.20, n=10).Whereas the second part are EWMA (L=2.814,λ=0.10, n=1), EWMA- ̅ (L=2.815,λ=0.10, n=10) and NPEWMA-SR (L=2.794,λ=0.10, n=10).The subgroup size of the NPEWMA-SR control chart is fixed at 10 as [14] suggested.[15] also mentioned that the minimum subgroup size needed to construct a traditional 3δ limits control chart in Wilcoxon Signed-Rank statistic.To avoid unfair comparison due to the difference in subgroup size between EWMA- ̅ and NPEWMA-SR control charts, subgroup size of 10 is fixed for EWMA- ̅ control chart.
The design parameters in the first and second parts of the control charts are used for testing the IC robustness and OOC performance.Seven different shape parameters each control chart in Weibull distribution (skewness level = 0.1, 0.5, 1.0, 2.0, 3.0, 4.0 and 5.0) and 8 levels of δ times standard deviation for shifting process mean (δ = 0, 0.25, 0.5, 0.75, 1, 1.5, 2 and 3).

IC Robustness
There are two options in choosing an acceptable ARL0 value, which are either ARL0 ≈ 370 or 500 ( [16], [17]).In this simulation study, combinations of design parameters to obtain the ARL0 ≈ 500 are arbitrarily chosen since larger ARL0 value indicates that there is less adjustment needed in a process without any shift in the process mean.Moreover, the robustness of the ARL0 in symmetrical and asymmetrical distributions is compared among the control charts.

OOC Performance
Whereas ARL1 value in the specific value of shift in the process mean between symmetrical and asymmetrical distributions are recorded to observe the OOC performance.Furthermore, the minimum sample required before the first OOC point in the NPEWMA-SR control chart is calculated by applying Equation 3.For example, when n=10, L=2.905, and λ=0.20, the control limits can be calculated based on Equation 3, where UCL/LCL= ±12.4596.Then, substitute the value for UCL/LCL into the second equation of Equation 3, minimum sample required before the first OOC point, i=1.1512 will be obtained.Since the first OOC point can only be an integer number, the NPEWMA-SR control chart can only signal the first out-of-control point on or beyond sample number 2.

Simulation Algorithm
Performance of the IC robustness and OOC detection between EWMA, EWMA- ̅ , and NPEWMA-SR control charts are compared.Comparisons will be conducted based on Weibull (λ, k) distributions with fixed scale parameter, λ=1 and varying shape parameter, k, and combined with varying process mean shift in δ times of standard deviation (δ =0, 0.25, 0.5, 0.75, 1, 1.5, 2 and 3).The shape parameter was increased to have the move for the Weibull distribution from symmetrical to become more asymmetrical distribution.The simulation steps are show as follows: Step 1: Specify the design parameter (λ, L) and subgroup size, n for parametric EWMA and NPEWMA-SR control charts, Step 2: Apply the specific design parametric for both parametric EWMA and NPEWMA-SR control charts in Normal distribution and Weibull distribution, Step 3: Generate 4000 observations for Normal (0,1) and Weibull (λ,k) with λ=1 and varying k, Step 4: Calculate the steady-state control limits using parametric EWMA and NPEWMA-SR control charts, Step 5: Record the number of subgroups needed until 1 st OOC point being detected in each control chart, Step 6: Repeat steps 1-5 for 2000 times, and Step 7: Use proc univariate of SAS 9.4 to obtain the ARL0 and ARL1 for each control chart.

RESULTS AND DISCUSSION
The ARL tables were displayed for EWMA, EWMA-X ̅ and NPEWMA-SR control charts under Normal and Weibull distributions and compare the IC robustness and OOC performance among the control charts.Under Normal (0,1) distribution for EWMA, EWMA- ̅ and NPEWMA-SR control charts with fixed λ value and varying L, the ARL value increases as the L increases from 2.6 to 3.1 and it showed decreasing trend when δ gets larger.Meanwhile, the OOC performance between these three control charts are also investigated.The first step is to set a design parameter (L, λ) for each control chart.In order to set the design parameters, EMWA (L=3.0,λ=0.25, n=1) with ARL0=482.594,EWMA- ̅ (L=3.1, λ=0.25, n=10) with ARL0=523.033and NPEWMA-SR (L=2.9, λ=0.20, n=10) control charts with ARL0=484.479were chosen for the comparison since the ARL0≈482.594.It was found that for the detection of OOC for δ = 0.25 to 3.0, the EWMA- ̅ control chart has smaller ARL1 value than both EWMA and NPEWMA-SR (Figure 1).More specifically, the OOC performance of NPEWMA-SR control chart is comparable to the EWMA control chart (Table 1).Table 2 shows that the ARL values increase as L gets larger, whereas ARL values decrease as the δ increases for all the control charts.Besides that, the design parameters for these control charts were chosen as EWMA (L=2.9, n=1), EWMA- ̅ (L=2.9, n=10) and NPEWMA (L=2.8, n=10) control charts with fixed λ=0.10 because the ARL0 is approximately equal to 500.By comparing ARL1 values on the Table 2 (and Figure 2), it showed that the EWMA- ̅ control charts have better OOC performance than EWMA and NPEWMA-SR control charts.Since the ARL1 values are not much different for EWMA- ̅ and NPEWMA-SR control charts indicated that the OOC performance is comparable between both control charts.Furthermore, the results also showed that different combinations of (L,λ) would not affect the ARL trend of the control chart when δ gets   By comparing the IC robustness for each control chart, the ARL0 value of the control charts showed decreasing trend when the distribution goes to more-asymmetrical (Table 3 and Figure 3 (a)).The results also indicated that the IC robustness of the control charts are poor when the symmetrical assumption is violated.The ARL0 values of EWMA- ̅ control chart were larger than both EWMA and NPEWMA-SR control charts.These indicate that the EWMA- ̅ control chart more robust for the IC in violation of symmetrical assumption than EWMA and NPEWMA-SR control charts.
Figure 3 (b) compared the control charts at δ = 0.25.The ARL1 value of EWMA control chart gets smaller until the shape parameter of Weibull distribution is equal to 0.6478 and slightly increases as the distribution becomes more asymmetric.However, the NPEWMA-SR shows an increasing trend for ARL1 value until the shape parameter equals 0.6478, and it decreases when the distribution becomes more asymmetric.In this study, it was found that the EWMA- ̅ control chart is the only control chart with few differences in ARL1 value when the distribution becomes more asymmetrical.It indicated that the violation in symmetrical assumption gives a bigger effect on small shift detection for NPEWMA-SR control chart than both parametric EWMA control charts.On the other hand, EWMA- ̅ control chart was the most stable control chart, which still able to detect slight shift in process mean.Besides that, it was found that the EWMA- ̅ control chart is the only OOC robust control chart for violation in symmetrical assumption since it shows nonfluctuated ARL1 values compared to EMWA and NPEWMA-SR control charts.
Unfortunately, the EWMA- ̅ control chart is not robust for the IC in the violation of the symmetrical assumption because the ARL0 value decreases when the distribution goes to asymmetrical.In violation of symmetrical assumption, NPEWMA-SR control chart loss its ability to detect small shift in process mean.However, it shown a comparable ARL1 value for OOC performance in δ = 0.5 and above.In short, these three control charts are unsuitable to apply in asymmetrical distribution, since false alarm will happen frequently.

CONCLUSIONS
The simulation study has shown that parametric EWMA and non-parametric EWMA control charts are adversely affected by symmetrical assumption violation.In symmetrical Weibull distribution, both control charts had an acceptable ARL0 value and OOC performance, which mean no frequent false alarm and fast detection in small shift of process mean.Unfortunately, both control charts were unsuitable for asymmetrical distribution due to unacceptable ARL0 value.Comparison among the control charts in symmetrical Weibull distribution showed that NPEWMA-SR control chart has a better small shift detection performance and comparable ARL0 value.In short, NPEWMA-SR control chart was one of the suitable methods to apply in nonnormal symmetrical distribution compared to parametric EWMA control chart.
In future work, modify parametric EWMA and NPEWMA-SR control charts by using the skewness corrective method to apply in asymmetrical distribution.Since the violation in symmetrical assumption will cause the non-parametric control chart to become unviable, a new non-parametric control chart with strong robustness in asymmetrical distribution will be considered for future study.

Figure 3 .
ARL Values for EWMA, EWMA- ̅ ), and NPEWMA-SR Control Chart by Holding Level of Shift in Weibull Distribution at λ=1 and Varying k: a) at a) δ = 0.00, b) δ = 0.25 Simulation Study for Parametric EWMA and NPWEWPA-SR Control Charts Against Non-Normality Assumptions Anwar Fitrianto 165

Table 1 .
Bias of Parameter Estimates of LD, MS and MI Method

Table 3 .
ARL Values under Weibull (λ,k) Distribution with Fixed λ=1 and Varying k Value for EWMA, EWMA- ̅ , and NPEWMA-SR Control Charts ARL1 values for NPEWMA-SR control chart converge to 2.000 due to the restriction of Equation 3.